A PRELIMINARY STUDY FOR A TETRAHEDRON FORMATION: QUALITY FACTORS AND VISUALIZATION Jos_ J. Guzm_n* Conrad SchitP a.i. solutions, Inc. Mission Design Division 10001 Derekwood Lane, Suite 215 Lanham, MD 20706, USA Abstract Spacecraft flying in tetrahedron formations are excellent for electromagnetic and plasma studies. The quality of the science recorded is strongly affected by the tetrahedron evolution. This paper is a preliminary study on the computation of quality factors and visualization for a formation of four or five satellites. Four of the satellites are arranged geometrically in a tetrahedron shape. If a fifth satellite is present, it is arbitrarily initialized at the geometric center of the tetrahedron. The fifth satellite could act as a collector or as a spare spacecraft. Tetrahedron natural coordinates are employed for the initialization. The natural orbit evolution is visualized in geocentric equatorial inertial and in geocentric solar magnetospheric coordinates. INTRODUCTION thc formation -- in coordinate frames useful to the scientists is important. This paper is a preliminary Spacecraft formation flying and/or distributed space study on the computation of quality .factors and visu- systems allow measurements not possible with single alization for a formation of four or five satellites. spacecraft. For space science missions, observable field parameters (particle populations, electric and mag- netic fields) vary both in space and time. L Thus, TETRAHEDRON MISSIONS understanding of the processes within the field re- Several missions are planning to use tetrahedron for- quires accurate and precise observations and analysis mations for field and plasma studies. In fact, one such of both the temporal and spatial variations. A tetra- mission, Cluster II*, is currently flying and operat- hedron formation is utilized since four spacecraft, with ing successfully. 3 The Cluster II mission was launched adjustable separations, arc the minimum needed to in pairs using Russian Soyuz rockets: two spacecraft resolve a three-dimensional structure, at least to the launched on July 12, 2000 and two on August 9, lowest order in the physical field gradients. 2 2000. Cluster's primary mission is to explore the Sun- The appropriate spacecraft separations, of course, Earth connection, specifically the interaction of the depend on the field or event being sampled. As a Sun and the Earth's magnetosphere. In the process result, formation flying missions require closer inter- of planning the Cluster mission, fuel optimization of action between the project scientists and mission de- the maneuvers needed to initialize, modify and main- signers than a typical single-spacecraft mission. A tain the formation was considered by J. Roddguez- means of communicating quantitatively is to define Canabal and M. Belld-Mora. 4 Later a different opti- quality factors that affect the data collection and anal- mization method was employed by J. Schoenmaekers. 5 ysis processes. These quality factors are shape- and In J. Schoenmaekers's paper, both methods are com- size-dependent parameters that are hmctions of the pared for one mission scenario. Moreover, a further- geometric shape and cvo]ution of the formation. Once developed strategy is presented by M. Belld-Mora and the scientists have defined the quality factors appro- Rodrfguez-Canabal. 6 Implementation and operational priate for their mission, the mission analysts can try results for the maneuvers are reported by D. Hockens to tailor the trajectory design appropriately. Further- and J. Sehoenmaekers. 7 more, to communicate qualitatively, visualization of *Cluster II is a replacement of the original Cluster mission *Mission Analyst, Member AIAA and AAS which was lost in a launch failure during the maiden flight of tChief Scientist,Member AAS the Ariane 5 rocket on June 4, 1996. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS The Auroral Lites mission concept also proposed us- where [RUm_'[ is the determinant of the tensor. Hence the name, volumetric tensor. ing four spacecraft. This mission's main goal would have been to acquire an unprecedented understanding Relationship to the Inertia Tensor of the multiscale dynamics of space plasmas by ex- The inertia tensor about the mesocenter is closely ploring the Earth's auroral zone. s'9 Auroral Lites also related to the volumetric tensor. In fact, the eigenvec- would have been a technology pathfinder for flying for- tots are the same and the eigenvalue associated with mations of small spacecraft. Unfortunately, the mis- the largest (smallest) eigenvalue of the inertia tensor sion did not make the final selection of the 1998 NASA is associated with the smallest (largest) eigenvalue of Medium-class Explorers (MIDEX) program. An- the volumetric tensor. See Harvey 1 for the equation other proposed mission that will utilize a tetrahedron relating the eigenvalues. formation is the Magnetospheric Multiscale (MMS) Mission. l° MMS will determine the small-scale basic Pseudo-Ellipsoid plasma processes which transport, accelerate and en- Analogous to the inertia-ellipsoid, a pseudo-ellipsoid ergize plasmas in thin boundary and current layers. can be constructed with the eigenvalues and eigenvec- These processes control the structure and dynamics of tors of the volumetric tensor. Therefore, the pseudo- the Earth's magnetosphere. Preliminary mission de- ellipsoid has information about tim shape and orienta- sign and analysis for the MMS reference path, which tion of the formation. This proves very useful in the includes a double lunar swingby for one of its phases, formulation of quality factors. has been presented by A. Edery and C. Schiff. n NATURAL COORDINATES THE VOLUMETRIC TENSOR For both mission design and science purposes, it The spatial gradient of a field observed with multiple might be convenient to sometimes work in a coordinate spacecraft is dependent on the inverse of a symmet- frame that is dependent on the shape of the tetrahe- ric tensor formed from the positions of the spacecraft dron. This can be accomplished by utilizing natural (see Harvey1). Harvey adds that "the importance of coordinates. Natural coordinates are typically used in the volumetric tensor for describing the geometry of finite element analysisJ 2 Analogous to the volumetric a tetrahedron was first noted by J. Schoenmaekers tensor, the tetrahedron natural coordinates are depen- of the European Space Operations Centre (ESOC) dent on the volume of the tetrahedron. A tetrahedron Flight Dynamics Division". Thus, this tensor is impor- in rectangular and natural coordinates is displayed in tant for both the gradient calculations and for providing Figure 1. The faces of the tetrahedron are numbered S/C 4 information about the shape and orientation of the for- mation. For N spacecraft, in an arbitrary frame I and Face2 (_2=0) _ _Fa_.l (_1=0) about the mesocenter (mean position or centroid), the volumetric tensor can be expressed as, N _l/mc 1 N-- E rm_ irrnc-:r i, (1) i=1 _] 1 / Face4 (_4=0) where the double bar is used to indicate a second order tensor, and _m_ is the position (column) vector of the i- y s/c 2 th spacecraft relative to the mesocenter, rnc. Relative to the origin of the arbitrary frame I, the mesocenter Fig. 1 Tetrahedron Natural Coordinates is computed as, such that the i-th face is opposite to the i-th vertex N 1 (spacecraft). The tetrahedral coordinates for an in- = Z (2) terior point, ( = {(1, _2, _3, _4} T, are defined as the i=l ratio of the volume of the subtetrahedron formed by where ri is the position vector of the i-th spacecraft the interior point and face i to the tetrahedron volume. relative to the origin of the arbitrary frame I. The That is, _ = {V1/V, V2/V, V3/V, Va/V} T. Thus, the mesocenter is equal to tile system's center of mass i-th face has _i = 0. Fklrthermore, note that if all the spacecraft have the same mass. Furthermore, if the tetrahedron is regular (the separations between vl + v2 + v_ + v4 = v, (4) vertices are equal), tile mesocenter equals the geo- which implies that, metric center (a point equidistant from all vertices). For a tetrahedron, it can be shown that the volume V _1+_2+(3+_4--1' (5) is given by, Therefore, with this last equation and relating the 8 V/_,_/m_l ' (3) cartesian coordinates {x, y, z} of an interior point to V=5 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS in defining the formation geometry and stationkeep- ing requirements. Before introducing some of them, various parameter definitions are required. -2190 Some Geometric Definitions -2191 -2192 -2189_ First, recall that in a regular tetrahedron the sep- -2193 _,-2194 arations between vertices are equal. Then, the ideal volume and surface are defined as the volume and -2197 surface area of a regular tetrahedron with vertex sepa- -2196 rations equal to the average of the 6 distances between -2199 the true, or actual, vertices. The circumscribing -4200 sphere is defined as the sphere with center at the ge- -2195_-4202 5158 ometric center and radius equal to the distance from -4204 5156 that center to any spacecraft (thus, all 4 spacecraft lie y _mJ 5150 x (Z_nl on its surface). One-Dimensional Parameters Fig. 2 Example: Inertial Coordinates The following parameters are useful for comparing the tetrahedral coordinates the true tetrahedron against an ideal (regular) tetra- hedron. A regular tetrahedron might be useful in cases 1 1 1 where sampling of the structure of a field is not as im- X Xl X2 X3 x4 _2 (6) portant as understanding its transient or fluctuating Y Yl Y2 Y3 Y4 _3 ' events. 13 {'}Z Zl Z2 Z3 Z41{_4 1} • The Glassmeier Q_M parameter is defined as, where the cartesian coordinates of the points that de- True Volume True Surface fine the tetrahedron are given by the vectors f_ = + + i. (7) QGM -- Ideal Volume Ideal Surface {xi, yi, zi} r for i = 1,..., 4.